\def \d{\mbox{d}}
\def \Exp#1{\mathbb{E}\left[#1\right]}
\def \Var#1{\mathbb{V}{\mbox{ar}}\left[#1\right]}
\def \Cov#1{\mathbb{C}{\mbox{ov}}\left(#1\right)}
\def \ft{\tilde{f}}
\def \fh{\hat{f}}
\def \bv{\boldsymbol{v}}
\def \bs#1{\boldsymbol{#1}}
\def \ExpQ#1{\mathbb{E}_\Q\left[#1\right]}
\def \Exph#1#2{\mathbb{E}^{#1}\left[#2\right]}

\def \vt{\vartheta}
\def \scr{\mathscr}
\def \vrho{\varrho}
\def \Vb{\overline{V}}
\def \unbr{\underbrace}

\def\und#1{\underline{#1}}
\def\ove#1{\overline{#1}}
\def\p{+}
\def\bb{\mathbb}

\def \Sum{\displaystyle\sum}
\def \Prod{\displaystyle\prod}
\def \Int{\displaystyle\int}
\def \Frac{\displaystyle\frac}
\def \Inf{\displaystyle\inf}
\def \Sup{\displaystyle\sup}
\def \Lim{\displaystyle\lim}
\def \Liminf{\displaystyle\liminf}
\def \Limsup{\displaystyle\limsup}
\def \Max{\displaystyle\max}
\def \Min{\displaystyle\min}

\def \be{\begin{eqnarray}}
\def \ee{\end{eqnarray}}
\def \b*{\begin{eqnarray*}}
\def \e*{\end{eqnarray*}}
\def \ba{\begin{array}}
\def \ea{\end{array}}

\def\no{\noindent}

\def \D{\mathbb{D}}
\def \E{\mathbb{E}}
\def \F{\mathbb{F}}
\def \G{\mathbb{G}}
\def \H{\mathbb{H}}
\def \I{\mathbb{I}}
\def \L{\mathbb{L}}
\def \M{\mathbb{M}}
\def \N{\mathbb{N}}
\def \P{\mathbb{P}}
\def \Q{\mathbb{Q}}
\def \R{\mathbb{R}}
\def \Z{\mathbb{Z}}


\def \[{[\,\!\![}
\def \]{]\,\!\!]}

\def \1{{\bf 1}}
\def \esssup{{\rm esssup}}

\def \proof{{\noindent \bf Proof. }}
\def \ep{\hbox{ }\hfill$\Box$}

\def \cl{{\rm cl}}
\def \interior{{\rm int}}

\def\reff#1{{\rm(\ref{#1})}}

%\addtolength{\oddsidemargin}{-0.1 \textwidth}
%\addtolength{\textwidth}{0.2 \textwidth}
%\addtolength{\topmargin}{-0.1 \textheight}
%\addtolength{\textheight}{0.2 \textheight}

\def\Ac{{\cal A}}
\def\Bc{{\cal B}}
\def\Cc{{\cal C}}
\def\Dc{{\cal D}}
\def\Ec{{\cal E}}
\def\Fc{{\cal F}}
\def\Gc{{\cal G}}
\def\Hc{{\cal H}}
\def\Jc{{\cal J}}
\def\Lc{{\cal L}}
\def\Mc{{\cal M}}
\def\Nc{{\cal N}}
\def\Kc{{\cal K}}
\def\k{\kappa}
\def\tK{\tilde K}
\def\tS{\tilde S}
\def\Lc{{\cal L}}
\def\Oc{{\cal O}}
\def\Pc{{\cal P}}
\def\Sc{{\cal S}}
\def\Tc{{\cal T}}
\def\Uc{{\cal U}}
\def\Vc{{\cal V}}
\def\Wc{{\cal W}}
\def\Xc{{\cal X}}
\def\Yc{{\cal Y}}
\def\Zc{{\cal Z}}

\def\vp{\varphi}
\def\indi{{\bf 1}}

\def\eps{\varepsilon}
 
 \def\vs#1{\vspace{#1mm}}
 \def\vv{\vs2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DEB DEF  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%% display %%%%%%%%%%%%%%%%%%%
\def \Sum{\displaystyle\sum}
\def \Prod{\displaystyle\prod}
\def \Int{\displaystyle\int}
\def \Frac{\displaystyle\frac}
\def \Inf{\displaystyle\inf}
\def \Sup{\displaystyle\sup}
\def \Lim{\displaystyle\lim}
\def \Liminf{\displaystyle\liminf}
\def \Limsup{\displaystyle\limsup}
\def \Max{\displaystyle\max}
\def \Min{\displaystyle\min}
\def\essinf{{\rm ess}\!\inf\limits}
\def\esssup{{\rm ess}\!\sup\limits}
%%%%%%%%%%%% bold %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\Cb{{\mathbf C}}
\def\Lb{{\mathbf L}}
\def \Hb{{\mathbf H}}
%%%%%%%%%%%%%%% mathbb %%%%%%%%%%%%%%%%%%%%%%%%%
\def \E{\mathbb{E}}
\def \F{\mathbb{F}}
\def \H{\mathbb{H}}
\def \M{\mathbb{M}}
\def \N{\mathbb{N}}
\def \R{\mathbb{R}}
\def \Z{\mathbb{Z}}
\def\P{\mathbb{P}}
\def\Q{\mathbb{Q}}
\def\G{\mathbb{G}}
\def\T{\mathbb{T}}
%%%%%%%%%%%%%%  bar %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\mub{{\bar \mu}}
\def\Xb{\bar X}
\def\Yb{\bar Y}
\def\Zb{\bar Z}
\def\Ub{\bar U}
\def\Gammab{\bar \Gamma}
%%%%%%%%%%%%%%% hat %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\Zh{{\hat Z}}
\def\Uh{{\hat U}}
\def\Gammah{\hat \Gamma}

%%%%%%%%%%%%%% caligraphic %%%%%%%%%%%%%%%%%%%%%
\def\Ac{{\cal A}}
\def\Bc{{\cal B}}
\def\Cc{{\cal C}}
\def\Dc{{\cal D}}
\def\Ec{{\cal E}}
\def\Fc{{\cal F}}
\def\Gc{{\cal G}}
\def\Hc{{\cal H}}
\def\Ic{{\cal I}}
\def\Jc{{\cal J}}
\def\Kc{{\cal K}}
\def\Lc{{\cal L}}
\def\Mc{{\cal M}}
\def\Nc{{\cal N}}
\def\Oc{{\cal O}}
\def\Pc{{\cal P}}
\def\Qc{{\cal Q}}
\def\Rc{{\cal R}}
\def\Sc{{\cal S}}
\def\Tc{{\cal T}}
\def\Uc{{\cal U}}
\def\Vc{{\cal V}}
\def\Wc{{\cal W}}
\def\Xc{{\cal X}}
\def\Yc{{\cal Y}}
\def\Zc{{\cal Z}}
%%%%%%%%%%%%% tilde %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\Xt{\tilde X}
\def\Yt{\tilde Y}
\def\Zt{\tilde Z}
%%%%%%%%%%%%% usual shortcuts %%%%%%%%%%%%%%%%%%
\def\ep{\hbox{ }\hfill$\Box$}
\def\reff#1{{\rm(\ref{#1})}}
%%%%%%%%%% tableaux %%%%%%%%%%%%%%%
\def\be{\begin{eqnarray}}
\def\ee{\end{eqnarray}}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\b*{\begin{eqnarray*}}
\def\e*{\end{eqnarray*}}
%%%%%%%% ensembles %%%%%%%%%%%%%%%%
\def\ri#1{\mbox{ri}(#1)}
\def\inte#1{\mbox{\rm int}(#1)}
\def\x{\times}
%%%%%% egalites conv %%%%%%%%%%%%%%
\def\eqlaw{=^{\hspace{-8mm} ^{\rm law}}}
\def \sge{\;\ge\;}
\def \sle{\;\le\;}
\def\={\;=\;}
\def\lra{\longrightarrow}
\def\conv{\mbox{\rm conv}}
%%%%%%%%% texte dans math et mise en forme %%%%%%%%%%%%%%%%%%%%
\def\proof{{\noindent \bf Proof. }}
\def\diag#1{\mbox{\rm diag}\left[#1\right]}
\def \ton{{\rm on }}
\def \tif{{\rm if }}
\def\And{\;\mbox{ and }\;}
\def\pourtout{\mbox{ for all } }
\def\Pas{\mathbb{P}-\mbox{a.s.}}
\def\no{\noindent}
\def\.{\;.}
\def\lb{\underline}
%%%%%%%% items %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\i{{\rm (i)}}
\def\ii{{\rm (ii)}}
\def\iii{{\rm (iii)}}
\def\iv{{\rm (iv)}}
%%%%%%% shotcuts lettres%%%%%%%%%%%%%%%%%%%%%%%%
\def\eps{\varepsilon}
\def\vp{\varphi}
\def\1{{\bf 1}}
%%%%%%%%%%%%%%%%%%%%   couleur %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\red#1{\textcolor[rgb]{1,0,0}{ #1}}
\def\green#1{\textcolor[rgb]{0,1,0}{ #1}}
\def\blue#1{\textcolor[rgb]{0,0,1}{ #1}}
%%%%%%%%%%%%%%%%% speciale %%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\IW#1#2#3{\left(#1\ast W\right)_{#2,#3}}
\def\Il#1#2#3{\left(#1\ast \ell\right)_{#2,#3}}
\def\Imu#1#2#3{\left(#1\ast \mu\right)_{#2,#3}}
\def\Inu#1#2#3{\left(#1\ast \nu\right)_{#2,#3}}
\def\Inub#1#2#3{\left(#1\ast \nub\right)_{#2,#3}}
\def\Hlambda{{\rm \bf H}_\lambda}
\def\D{{{\rm I\hspace{-0.5mm}D}}}
\def\Dd{{{\rm I\hspace{-0.5mm}D}^{2,2}}}
\def\ckp{C_{p}\;}
\def\ck{C_{2}\;}
\def\ckX{C^0_{2}\;}

 
%%%%%%%%%%%%% pi %%%%%%%%%%%%%%%%
\def\Xp{X^\pi}
\def\Yp{Y^\pi}
\def\Zp{Z^\pi}
\def\Up{U^\pi}
\def\Gammap{\Gamma^\pi}

\def\Ypb{\bar{Y}^\pi}
\def\Zpb{\bar{Z}^\pi}
\def\Upb{\bar{U}^\pi}

\def\Yhp{\hat Y^{\pi}}
\def\Zhp{\hat Z^{\pi}} 
\def\Gammapb{\bar{\Gamma}^\pi}
\def\br{{\bf r}}
\def\CE{{\text{CE}}}
\def\CA{{\text{CA}}}

%%%%%%%%%%%%%%%% operateur math proba%%%%%%%%%%%
\def\Esp#1{\mathbb{E}\left[#1\right]}
\def\Espt#1{\mathbb{E}^{\tilde \P}\left[#1\right]}
\def\Esph#1{\hat{\mathbb{E}}\left[#1\right]}
\def\Espp#1{\mathbb{E}^{\P}\left[#1\right]}
\def\Et#1{\tilde{\mathbb{E}}\left[#1\right]}

\newcommand\Eu[2]{\mathbb{E}_{#1}\left[#2\right]}

\def\Var#1{{\rm Var}\left[#1\right]}
\def\Pro#1{\mathbb{P}\left[{#1}\right]}
\def\mod#1{\left|#1\right|}
\def\scap#1#2{\langle #1 , #2\rangle} 
\def\einf#1{\mathop{{\rm ess\!-\!inf}}_{#1}\;}
\def\esup#1{\mathop{{\rm ess\!-\!sup}}_{#1}\;}
\def\argmax{\mbox{\rm arg}\max}
\def\Tr#1{\mbox{\rm Tr}\left[#1\right]}
\newcommand\Ex[3]{\mathbb{E}_{#1}^{#2}\left[#3\right]}

 %%%%%%%%%%%%%%%% arbitrage %%%%%%%%%%%%%%%%%%%%%
\def\NA{{\bf NA }}
\def\HKP{\mbox{\rm{\bf KP}}}
\def\HNF{\mbox{\rm{\bf HN}}}
\def\HFa{\mbox{\rm{\bf HF$_1$}}}
\def\HFb{\mbox{\rm{\bf HF$_2$}}}
\def\HL{\mbox{\rm{\bf H$\Oc$}}}
\def\NAw{\mbox{\rm{\bf NA$^w$}}}
\def\NAs{\mbox{\rm{\bf NA$^s$}}}
\def\NAr{\mbox{\rm{\bf NA$^r$}}}

%%%%%%%%%%%%%%%%% bipolarite %%%%%%%%%%%%%%%%%%
\def\EcondG#1{\mathbb{E}\left[{#1}~|~{\cal G}\right]}
\def\po#1{#1{}^{(\Gc\!,\;o)}}
\def\bpo#1{#1{}^{(\Gc\!,\;oo)}}
\def\pos#1{#1{}^{o}}
\def\bpos#1{#1{}^{oo}}
\def\dbover#1{\overline{{#1}}}
\def\dboverG#1{{\overline{#1}^{\Gc}}}

%%%%%%%% options barriere %%%%%%%%%%%%%%ù
\def\Hsigma{{\bf H$_{\sigma}$}}
\def\HK{{${\rm \bf H}_{K}$}}
\def\Hg{{\bf H$_{g}$}}
\def\HD{{\bf H$_{D}$}}
\def\HO{{\bf H$_{\Oc}$}}
\def\HKt{{\bf H$_{\tilde K}$}}
\def\tx{^{t,x}}
\def\utx{_{_{t,x}}}

%%%%%%%%%%% E condi et BSDE%%%%%%%%%%%%%%%%%%
\def \BCD{\texttt{BCD}}
\def\Ehp{\hat \E^{\pi}}
\def\Ep{\E^{\pi}}

%%%%%%%%%%%%%%% dates et schema discretisations%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\ti{{t_i}}
\def\tip{ {t_{i+1}} }
\def\tim{ {t_{i-1}}}
\def\tauip{ {\tau_{i+1}} }
\def\taui{{\tau_{i}}}
\def\tauim{{\tau_{i-1}}}
\def\ip{{{i+1}}}
\def\im{{{i-1}}}
\def\DW{\Delta W}




\def\tha{{\tau_{h^\alpha}^N}}
\def\tax{{\tau_{\cA}^{t_0,x}}}
\def\ta{{\tau_{\cA}}}
\def\thb{\tau_{h^\beta}^N}
\def\H{{\mathbb{H}}}
\def\sS{{\mathbb{S}}}
\def\dO{{\partial{\cal O}}}
\def\cG{{\mathcal{G}}}
\def\cD{{\mathcal{D}}}
\def\cK{{\mathcal{K}}}
\def\cC{{\mathcal{C}}}
\def\cO{{\mathcal{O}}}
\def\cA{{\mathcal{A}}}
\def\cN{{\mathcal{N}}}
\def\cU{{\mathcal{U}}}
\def\cP{{\mathcal{P}}}
\def\ch{{\rm{ch}}}
\def\argch{{\rm{argch}}}
\def\argsh{{\rm{argsh}}}
\def\sh{{\rm{sh}}}
\def\ue{{u_\varepsilon}}
\def\Xe{X^\varepsilon}
\def\Ye{Y^\varepsilon}
\def\tYe{\tilde Y^\varepsilon}
\def\Ze{Z^\varepsilon}
\def\tZe{\tilde Z^\varepsilon}
\def\te{\tau^\varepsilon}
\def\leftB{[\![}
\def\rightB{]\!]}